Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L2 (Rd )

Jean Michel Combes; Peter D. Hislop

doi:10.1007/978-3-030-68490-7_6

Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d )

Jean Michel Combes, Peter D. Hislop

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes ⊂ R^d . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes ⊂ R^d, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

Original language	English
Title of host publication	Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev
Editors	Sergio Albeverio, Anindita Balslev, Ricardo Weder
Pages	117-132
Number of pages	16
DOIs	https://doi.org/10.1007/978-3-030-68490-7_6
State	Published - 2021
Event	Conference to celebrate Erik Balslev’s 75th birthday, 2010 - Aarhus, Denmark Duration: Oct 1 2010 → Oct 2 2010

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	348
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	Conference to celebrate Erik Balslev’s 75th birthday, 2010
Country/Territory	Denmark
City	Aarhus
Period	10/1/10 → 10/2/10

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021.

Keywords

Density of states
Random schrödinger operators
Spectral averaging

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-030-68490-7_6

Cite this

Combes, J. M., & Hislop, P. D. (2021). Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). In S. Albeverio, A. Balslev, & R. Weder (Eds.), Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev (pp. 117-132). (Springer Proceedings in Mathematics and Statistics; Vol. 348). https://doi.org/10.1007/978-3-030-68490-7_6

Combes, Jean Michel ; Hislop, Peter D. / Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. editor / Sergio Albeverio ; Anindita Balslev ; Ricardo Weder. 2021. pp. 117-132 (Springer Proceedings in Mathematics and Statistics).

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abstract = "We prove some local estimates on the trace of spectral projectors for random Schr{\"o}dinger operators restricted to cubes ⊂ Rd . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schr{\"o}dinger operators on cubes ⊂ Rd, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.",

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Combes, JM & Hislop, PD 2021, Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). in S Albeverio, A Balslev & R Weder (eds), Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. Springer Proceedings in Mathematics and Statistics, vol. 348, pp. 117-132, Conference to celebrate Erik Balslev’s 75th birthday, 2010, Aarhus, Denmark, 10/1/10. https://doi.org/10.1007/978-3-030-68490-7_6

Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). / Combes, Jean Michel; Hislop, Peter D.
Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. ed. / Sergio Albeverio; Anindita Balslev; Ricardo Weder. 2021. p. 117-132 (Springer Proceedings in Mathematics and Statistics; Vol. 348).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes ⊂ Rd . We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes ⊂ Rd, for d 1. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

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Combes JM, Hislop PD. Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L² (R^d ). In Albeverio S, Balslev A, Weder R, editors, Schrödinger Operators, Spectral Analysis and Number Theory - In Memory of Erik Balslev. 2021. p. 117-132. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-030-68490-7_6